115

Documenta Math.

The Integral Cohomology Algebras of Ordered Configuration Spaces of Spheres ¨nter M. Ziegler Eva Maria Feichtner and Gu

Received: August 12, 1999 Communicated by Ulf Rehmann

Abstract. We compute the cohomology algebras of spaces of ordered point configurations on spheres, F (S k , n), with integer coefficients. For k = 2 we describe a product structure that splits F (S 2 , n) into well-studied spaces. For k > 2 we analyze the spectral sequence associated to a classical fiber map on the configuration space. In both cases we obtain a complete and explicit description of the integer cohomology algebra of F (S k , n) in terms of generators, relations and linear bases. There is 2-torsion occuring if and only if k is even. We explain this phenomenon by relating it to the Euler classes of spheres. Our rather classical methods uncover combinatorial structures at the core of the problem. 2000 Mathematics Subject Classification: Primary 55M99; Secondary: 57N65, 55R20, 52C35 Keywords and Phrases: spheres, ordered configuration spaces, subspace arrangements, integral cohomology algebra, fibration, Serre spectral sequence

1

Introduction

The space of configurations of n pairwise distinct labelled points in a topological space X, F (X, n) := {(x1 , . . . , xn ) ∈ X n | xi 6= xj for i 6= j} ⊆ X n , is called the n-th (ordered) configuration space of X. A systematic study of these spaces started with work by Fadell & Neuwirth [FaN] and Fadell [Fa] in the sixties. They introduced sequences of Documenta Mathematica 5 (2000) 115–139

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fibrations for configuration spaces and mainly concentrated on describing their homotopy groups for various instances of X. In 1969 Arnol0 d [Ar] derived the integer cohomology algebra of F (C, n) — the group cohomology of the colored braid group — and thereby initiated still ongoing research on the cohomology algebras of complements of linear subspace arrangements. Broader interest in the cohomology algebras of configuration spaces came up in the seventies: The cohomology of F (X, n) for a manifold X appeared as a basic ingredient in the E2 -terms of spectral sequences for the Gelfand-Fuks cohomology of the manifold [GF] and for the homology of certain function spaces [An]. Cohen [C1, C2] studied various aspects of the cohomology of configuration spaces of Euclidean spaces in view of its relation to homology operations for iterated loop spaces [C3]. Cohen & Taylor [CT1, CT2] described the cohomology algebras of configuration spaces of spheres with coefficients in a field of characteristic different from 2. Recently, compactifications of configuration spaces of algebraic varieties have been constructed by Fulton and MacPherson [FM]. As an application, they determine the rational homotopy type of configuration spaces of non-singular compact complex algebraic varieties F (X, n) in terms of invariants of X. Compare also work of Kriz [Kr] and Totaro [T], where alternative minimal models for F (X, n) are used. In contrast to these results on the rational homotopy type of configuration spaces, it seems that so far Arnol0 d’s computation of the integer cohomology algebra of F (C, n) remained the only instance where the integer cohomology algebra of an ordered configuration space was fully described. Recently, Raoul Bott asked about the integer cohomology algebra of the ordered configuration space of the 2-sphere. We are able to answer his question by describing a product decomposition for F (S 2 , n): F (S 2 , n)

∼ = PSL(2, C) × M0,n ,

where M0,n , the moduli space of n-punctured complex projective lines, is homotopy equivalent to the complement of an affine complex hyperplane arrangement. We deduce that H ∗ (F (S 2 , n), Z) has (only) 2-torsion that can be traced back to H 2 (PSL(2, C), Z) ∼ = Z2 (Section 2). For spheres of higher dimension we use spectral sequences to obtain an analogous decomposition on the level of cohomology algebras: H ∗ (F (S k , n), Z) H ∗ (F (S k , n), Z)

(k) ∼ = (Z ⊕ Z) ⊗ H ∗ (M(An−2 ), Z) ∼ = (Z ⊕ Z2 ⊕ Z) ⊗ H ∗ (M(AΠ3 ), Z)

(k)

for odd k , for even k ,

where M(An−2 ) is the complement of a certain arrangement of real linear (k) subspaces An−2 and M(AΠ3 ) is the complement of an arrangement of affine (k) subspaces that is naturally related to the linear arrangement An−2 . For both arrangement complements the integer cohomology algebra is torsion-free and we have explicit descriptions in terms of generators, relations and linear bases. In the following all (co)homology is taken with Z-coefficients. Documenta Mathematica 5 (2000) 115–139

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The key for our approach is a family of locally trivial fiber maps on configuration spaces that appears already in the work by Fadell & Neuwirth [FaN] and Fadell [Fa]. The maps are given by “projection to the last r points” of a configuration. For configuration spaces of spheres F (S k , n) and 1 ≤ r < n the projection Πr reads as follows: Πr = Πr (S k , n) :

F (S k , n) −→ F (S k , r) (x1 , . . . , xn ) 7−→ (xn−r+1 , . . . , xn ) .

We derive the integer cohomology algebra of F (S k , n) for k > 2 by a complete discussion of the Leray-Serre spectral sequence associated to the fiber map Π1 (S k , n). Our success with this rather classical approach depends on the fact that the fibers of Π1 (S k , n) are complements of linear subspace arrangements. Their cohomology algebras are well-studied objects both from topological and combinatorial viewpoints [GM, BZ, Bj, DP]. The fibers of Π1 (S k , n) are in fact the complements of codimension k versions of the classical braid arrangements, and thus they are particularly prominent examples of arrangement complements. This paves the way for a complete discussion of the associated spectral sequence (Section 3). A distinction between the configuration spaces of spheres of odd and even dimension emerges from the only possibly non-trivial differential of the spectral sequence. We present two methods to compute this differential (Section 4). (1) It can be derived from one particular cohomology group of F (S k , n). To obtain the latter we use an independent, rather elementary approach to the cohomology of configuration spaces, which may be of interest on its own right. (2) We show that the differential can be interpreted as a map that is induced by “multiplication with the Euler class of S k .” It is well-known that the Euler class depends on the parity of k. To get the final tableau of the spectral sequence, and to derive the integer cohomology algebra of the configuration space F (S k , n), we use combinatorially constructed Z-linear bases for the cohomology of the fiber (Section 5). In the last section of this paper we consider the bundle structures on F (S k , n) given by the fiber maps Πr (S k , n), 1 < r < n. We show that the associated spectral sequences collapse in their second terms unless k is even and r equals 1 or 2. For some parameters we can decide the triviality of the bundle structure, which in general is a difficult question. For configuration spaces of closed manifolds other than spheres, in principle one can attempt to follow the approach taken in this paper. However, with the cohomology of the manifold (i.e., of the base space of the considered fiber map) getting more complicated, the corresponding spectral sequence will be less sparse, and thus more non-trivial differentials will have to be considered. Even more importantly, if the manifold is not simply connected, then it is not straightforward, and not true in general, that the system of local coefficients Documenta Mathematica 5 (2000) 115–139

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on the manifold induced by the fiber map is simple. Already the entries of the second sequence tableau thus will be much harder to compute. Acknowledgment: We are grateful for discussions with Ezra Getzler that influenced the course of these investigations. Also, we wish to thank Raoul Bott who asked us about connections to the Euler classes of spheres. 2

Configuration spaces of the 2-sphere

We first comment on some special cases for small values of n and on the configuration space of the 1-sphere. For n = 1, we see from the definition that F (X, 1) = X for all spaces X. For n = 2, we consider the projection Π1 , sending a configuration in F (S k , 2) to its second point. We obtain a fiber bundle k k k ∼ k with contractible fiber Π−1 1 (x2 ) = F (S \{x2 }, 1) = R , hence F (S , 2) ' S . k k In fact, F (S , 2) is equivalent to the tangent bundle over S . For the configuration space of the 1-sphere, F (S 1 , n), we state an explicit trivialization of the fiber bundle given by Π1 , the projection to the last point of a configuration. Using the group structure on S 1 we define a homeomorphism which shows that Π1 (S 1 , n) is a trivial fiber map: ϕ1 :

F (S 1 \ {e}, n − 1) × S 1 −→ F (S 1 , n) ((x1 , . . . , xn−1 ) , y) − 7 → (yx1 , . . . , yxn−1 , y) .

For r > 1, the fiber of Πr (S 1 , n) is homeomorphic to the space of configurations of n − r points on r disjoint copies of the unit interval. We obtain a homeomorphism ϕr :

F(

U

r

(0, 1) , n − r) × F (S 1 , r)

−→ F (S 1 , n)

U that trivializes the bundle by “inserting” the points x1 , . . . , xn−r from r (0, 1) into the r open segments in which the points of the configuration (y1 , . . . , yr ) in F (S 1 , r) separate S 1 . Compared to configuration spaces of higher dimensional spheres we gain the main structural advantage for the 2-dimensional case from the fact that the 2-sphere S 2 is homeomorphic to the complex projective line CP 1 . We will freely switch between the resulting two viewpoints on the configuration space in question. The group of projective automorphisms PSL(2, C) of CP 1 acts freely on the configuration space F (CP 1 , n) by coordinatewise action, thus exhibiting F (CP 1 , n) as the total space of a principal PSL(2, C)-bundle for n ≥ 3 [Ge]. We identify the base space — the space of n-tuples of distinct points on the complex projective line modulo projective automorphisms — as the moduli space M0,n of n-punctured complex projective lines. Compactifications of M0,n and their cohomology algebras are the focus of recent research; for a brief account and further references see [FM, p.189]. Documenta Mathematica 5 (2000) 115–139

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Theorem 2.1 The configuration space F (CP 1 , n) of the complex projective line is the total space of a trivial PSL(2, C)-bundle over M0,n for n ≥ 3; hence there is a homeomorphism F (CP 1 , n) ∼ = PSL(2, C) × M0,n . Proof. The automorphism group PSL(2, C) acts sharply 3-transitive on CP 1 . In particular, we obtain a homeomorphism between the configuration space of three distinct points on CP 1 and the automorphism group PSL(2, C): φ : F (CP 1 , 3) −→ PSL(2, C) . Here (x1 , x2 , x3 ) ∈
F (CP 1 , 3)
is mapped to
the unique automorphism that transforms x1 to 10 , x2 to 01 , and x3 to 11 , i.e., to the “standard projective basis” of CP 1 . Given a configuration x = (x1 , . . . , xn ) of n distinct points on CP 1 , the group element φ(x1 , x2 , x3 ) transforms x to a configuration on CP 1 that has the standard projective basis in its first three entries. We describe the resulting configuration by the columns of a (2 × n)-matrix:   1 0 1 z3 . . . zn−1 φ(x1 , x2 , x3 ) ◦ x = , 0 1 1 1 ... 1 where zi ∈ C\{0, 1} for 3 ≤ i ≤ n − 1, zi 6= zj for 3 ≤ i < j ≤ n − 1, and the columns are understood as vectors in C2 \{0} that represent elements in CP 1 . Lifting an element x ¯ ∈ M0,n to its “normal form” φ(x1 , x2 , x3 ) ◦ x in the total space F (CP 1 , n) defines a section for the PSL(2, C)-bundle. Hence, the principal bundle is trivial [St, Part I, Thm. 8.3]. The resulting product decomposition on F (CP 1 , n) can be described explicitly by the homeomorphism Φ :

F (CP 1 , n) −→ PSL(2, C) × M0,n (x1 , . . . , xn ) 7−→ ( φ(x1 , x2 , x3 ) , x ¯).

2

4

Remark 2.2 An analogous argument is not possible for S , since there are no sharply 3-transitive group actions in the case of a non-commutative field such as H. The structural reason for this can be traced back to a theorem by von Staudt, see [P, Kap. 5.1.4]. In view of a description of the integer cohomology algebra of F (CP 1 , n) we use the intimate relation of the base space M0,n to a complex hyperplane arrangen−1 ment — the complex braid arrangement AC given by n−2 of rank n − 2 in C the hyperplanes zj − z i = 0 for 1 ≤ i < j ≤ n − 1 . This arrangement is a key example in the theory of hyperplane arrangements and initiated S much of its development [Ar, OT]. Its complement, n−1 M(AC ) := C \ AC n−2 n−2 , coincides with F (C, n − 1), the configuration space of the complex plane. Documenta Mathematica 5 (2000) 115–139

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The base space M0,n is homotopy equivalent to the complement of the affine C arrangement affAC n−2 , which is obtained from An−2 by restriction to the affine n−2 hyperplane {z2 − z1 = 1} ∼ . A complete description the integer = C S of n−2 aff C cohomology algebra of the complement M(affAC ) := C \ A n−2 n−2 is provided by general theory on the topology of complex hyperplane arrangements [OS, BZ, OT]. The description depends only on combinatorial data of the arrangement, i.e., on the semi-lattice of intersections L(affAC n−2 ) which is customarily ordered by reverse inclusion. Proposition 2.3 The base space M0,n is homotopy equivalent to the complement of the affine complex braid arrangement of rank n − 2, since M0,n × C ∼ = M(affAC n−2 ) . Its integer cohomology algebra is torsion-free. It is generated by one-dimensional classes ei,j for 1 ≤ i < j ≤ n − 1, (i, j) 6= (1, 2), and has a presentation as a quotient of the exterior algebra on these generators: ∼ ∗ (n−1 2 )−1 / I , H ∗ (M(affAC n−2 )) = Λ Z where I is the ideal generated by elements of the form ei,l ∧ ej,l − ei,j ∧ ej,l + ei,j ∧ ei,l

for

1 ≤ i < j < l ≤ n − 1, (i, j) 6= (1, 2) ,

e1,i ∧ e2,i

for

2 < i ≤ n− 1.

Proof. We consider the homeomorphic image of M0,n under the section defined in the proof of Proposition 2.1: ( )  1 0 1 z3 . . . zn−1 ∼ M0,n = zi ∈ C\{0, 1}, zi 6= zj for i 6= j 0 1 1 1 ... 1 ∼ = { (z1 , . . . , zn−1 ) | zi ∈ C, zi 6= zj for i 6= j, z1 = 0, z2 − z1 = 1} .

From this description we see that M0,n is homeomorphic to the complement of the affine braid arrangement affAC n−2 intersected with the hyperplane {z1 = 0}. This intersection operation is equivalent T toC a projection parallel to the intersection of all the hyperplanes in AC , An−2 = {z1 = . . . = zn−1 n−2 T}. The fibers of this projection map are contractible: they are translates of AC n−2 . Hence the projection does not alter the homotopy type, and we conclude that M0,n is homotopy equivalent to M(affAC n−2 ). The presentation of the integer cohomology algebra follows from general results on the topology of the complements of complex hyperplane arrangements (compare [OT]). 2 We have seen that the fiber PSL(2, C) is homeomorphic to F (CP 1 , 3), resp. F (S 2 , 3). By a result of Fadell [Fa, Thm. 2.4] there is a fiber homotopy Documenta Mathematica 5 (2000) 115–139

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equivalence between F (S k , 3) and Vk+1,2 , the Stiefel manifold of orthogonal 2-frames in Rk+1 . The cohomology of the latter is well-known, see [Bd, Ch. IV, Exp. 13.5]. Combining the product structure on F (CP 1 , n) obtained in Theorem 2.1 with the information on the cohomology algebras of base space and fiber we conclude: Theorem 2.4 The cohomology algebra of F (S 2 , n) with integer coefficients is given by H ∗ (F (S 2 , n))

∼ = ∼ =

H ∗ (F (S 2 , 3)) ⊗ H ∗ (M(affAC n−2 )) M
∗ Z(0) ⊕ Z2 (2) ⊕ Z(3) ⊗ Λ Z(1) / I , n−1 ( 2 )−1

where G(i) denotes a direct summand G in dimension i, and I is the ideal of relations described in Proposition 2.3. 3

A spectral sequence for H ∗ (F (S k , n))

Our approach for k > 2 uses the Leray-Serre spectral sequence associated with the projection Π1 : Π1 :

F (S k , n) −→ S k (x1 , . . . , xn ) 7−→ xn .

For the construction and special features of Leray-Serre spectral sequences we refer to Borel [Bo2, Sect. 2]. Since the base space of the considered fiber bundle is a sphere we could equally work with the Wang sequence [Wh, Ch. VII, Sect. 3], a long exact sequence connecting the cohomology of the total space and of the fiber. However, the derivation of the multiplicative structure of the cohomology algebra gets more transparent with spectral sequence tableaux. Moreover, this approach extends to projections Πr for r > 1 (see Section 6). We meet especially favorable conditions in the second tableau of the Leray-Serre spectral sequence associated to the fiber map Π1 (S k , n): The base space S k is simply connected for k ≥ 2, hence the system of local coefficients on S k induced by Π1 for k ≥ 2 is simple. As the fiber over xn ∈ S k we obtain: k n−1 Π−1 | xi 6= xj for i 6= j, 1 (xn ) = {(x1 , . . . , xn−1 ) ∈ (S )

∼ =

k n−1

{(x1 , . . . , xn−1 ) ∈ (R )

xi 6= xn for i = 1, . . . , n−1} | xi 6= xj for i 6= j} . (k)

This is the complement of the real k-braid arrangement An−2 of rank n−2 which is formed by linear subspaces Ui,j in (Rk )n−1 , 1 ≤ i < j ≤ n−1, Ui,j

= {(x1 , . . . , xn−1 ) ∈ (Rk )n−1 | xi1 = xj1 , . . . , xik = xjk } . Documenta Mathematica 5 (2000) 115–139

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This arrangement, a direct generalization of the real and complex braid arrangements, is a k-arrangement in the sense of Goresky & MacPherson [GM, Part III, p. 239]: the subspaces have codimension k, and the codimensions of their intersections are multiples of k. Such arrangements have combinatorial properties analogous to those of complex hyperplane arrangements, which is reflected by strong similarities in their topological properties: The cohomology algebras of real k-arrangements are torsion-free [GM, Part III, Thm. B]; they are generated in dimension k − 1 by cohomology classes that naturally correspond to the subspaces of the arrangement [BZ, Sect. 9]. (k) The complement of the real k-braid arrangement An−2 is an ordered configuration space: the space F (Rk , n − 1) of configurations of n − 1 pairwise distinct points in Rk . The following thus complements work by Cohen [C1, C2], who discussed the cohomology of F (Rk , n − 1) in connection with homology operations for iterated loop spaces. (k)

Proposition 3.1 The integer cohomology algebra of M(An−2 ) is generated by (k − 1)-dimensional cohomology classes ci,j , 1 ≤ i < j ≤ n − 1. It has a presentation as a quotient of the exterior algebra on these generators: n−1 (k) H ∗ (M(An−2 )) ∼ = Λ∗ Z( 2 ) / I ,

where I is the ideal generated by the elements (ci,l ∧cj,l ) + (−1)k+1 (ci,j ∧cj,l ) + (ci,j ∧ci,l )

for 1 ≤ i < j < l ≤ n−1 .

Remark 3.2 The generating cohomology classes ci,j , 1 ≤ i < j ≤ n−1, are defined by restricting cohomology generators b ci,j for the subspace complements M({Ui,j }) ' S k−1 to the complement of the arrangement. A canonical choice of the generators b ci,j results from fixing the natural “frame of hyperplanes” in the sense of [BZ, Sect. 9]. ¨ rner & Ziegler [BZ, Sect. 9] derived a presentation for the Proof. Bjo cohomology algebras of real k-arrangements up to the signs in the relations. For the real k-braid arrangement their presentation specializes up to signs to the one stated above. Consider the relation for a triple 1 ≤ i < j < l ≤ n − 1: ε1 (ci,l ∧ cj,l ) + ε2 (ci,j ∧ cj,l ) + ε3 (ci,j ∧ ci,l ) = 0 , εr ∈ {±1} for r = 1, 2, 3 . Transpositions of (i, j) and (i, l) and of (i, l) and (j, l) in the linear (lexico(k) graphic) order of the subspaces in An−2 lead to similar relations among the cohomology classes ci,l ∧ cj,l , ci,j ∧ cj,l , and ci,j ∧ ci,l : ε1 (ci,j ∧ cj,l ) + ε2 (ci,l ∧ cj,l ) + ε3 (ci,l ∧ ci,j ) = 0 ε1 (cj,l ∧ ci,l ) + ε2 (ci,j ∧ ci,l ) + ε3 (ci,j ∧ cj,l ) = 0 . Anti-commutativity of the exterior product yields the signs in the relations. 2 Documenta Mathematica 5 (2000) 115–139

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We obtain the following tensor product decomposition on the E 2 -tableau of the Leray-Serre spectral sequence associated with the fiber map Π1 (S k , n): E2∗,∗

3k − 3

dk (k)

H ∗ (M(An−2 ))

2k − 2

k−1 (k) E2p,q ∼ = H p (S k ) ⊗ H q (M(An−2 )) , p, q ≥ 0 .

0 0

k

The location of non-zero entries shows that there is only one possibly non-trivial differential on stage k of the sequence. 4

The k-th differential

The tableaux of a cohomological spectral sequence are bigraded algebras. The differentials respect their multiplicative structure. In particular, the differentials are determined by their action on multiplicative generators of the sequence tableaux. Thus, it suffices in our case to describe dk on the multiplicative gen(k) erators ci,j , 1 ≤ i < j ≤ n − 1, of Ek0,∗ ∼ = H ∗ (M(An−2 )) in dimension k − 1. Actually, we can restrict our attention even further to the action of dk on one single generator, say on c1,2 : The permutation of the first n − 1 points of a configuration in F (S k , n) by Sn−1 gives a group action on the considered fiber bundle and hence induces a Sn−1 -action on the spectral sequence. The group Sn−1 acts transitively on the generators ci,j of Ek0,k−1 , whereas it keeps Ekk,0 fixed. We conclude that dk (ci,j ) = dk (c1,2 )

for 1 ≤ i < j ≤ n − 1 .

In the following we provide two independent ways to evaluate dk . 4.1

. . . via a homology group of the discriminant.

Here the key observation is that knowing H k (F (S k , n)) is sufficient to determine dk . To obtain this specific group, we use a “Vassiliev type” argument that allows one to compute, in favorable situations, some cohomology groups of configuration spaces. Using a smooth compactification, in our case given by F (S k , n) ⊆ (S k )n , we set F (S k , n) = (S k )n \ Γn = (S k )n \

[

(Γn )i,j ,

1≤i 2, is given by  Z for odd k , k k ∼ H (F (S , n)) = Z2 for even k . Remark 4.3 In principle, the discriminant approach can be used to determine the cohomology of F (S k , n) as a graded group. However, to compute H∗ (Γn ) is difficult and requires extra tools (interpretation of Γn as a homotopy limit of a diagram of spaces, study of a spectral sequence converging to the homology ˇ Sect. 3(e)]). Also, the study of the pair sequence gets of a homotopy limit [ZZ, considerably more involved. Moreover, because of the use of Poincar´e-Lefschetz duality the multiplicative structure of H ∗ (F (S k , n)) seems out of reach for this approach. The partial result of Proposition 4.2 allows us to determine the differential in the spectral sequence associated to Π1 (S k , n). Taking cohomology of Ek∗,∗ with ∗,∗ respect to the differential dk leads to the final sequence tableau Ek+1 : Ek∗,∗ k−1

ci,j

0 k−1

∗,∗ Ek+1

ker dk

dk ν

0 0

coker dk

0 0

k

k

H k (F (S k , n)) Since there is only one non-zero entry on the k-th diagonal for k > 2, ∗,∗ H k (F (S k , n)) can be read from Ek+1 : H k (F (S k , n)) ∼ = coker dk . Our result on H k (F (S k , n)) in Proposition 4.2 implies that dk (c1,2 ) = dk (ci,j ) =



0 2ν

for odd k for even k ,

where ν is a generator of H k (S k ). Documenta Mathematica 5 (2000) 115–139

128 4.2

¨nter M. Ziegler Eva Maria Feichtner and Gu . . . via an interpretation in terms of the Euler class.

Our second approach to the differential dk stays within the setting of fiber bundles. We study an inclusion of fiber bundles and transfer information on the differentials via the induced homomorphism of spectral sequences. We will find that the differential is determined by the Euler class of the base space S k , which depends on the parity of k. Consider, for n ≥ 3, the following space of point configurations on S k , k > 2: Fb := {(x1 , . . . , xn ) ∈ (S k )n | x1 6= x2 , xj 6= xn for j = 1, . . . , n − 1} .

b : Fb → S k , makes it the total Projection of a configuration to its last point, Π space of a fiber bundle with spherical fiber: the complement of the codimension k subspace U1,2 in (Rk )n−1 , b −1 (xn ) = {(x1 , . . . , xn−1 ) ∈ (S k )n−1 | x1 6= x2 , xj 6= xn for 1 ≤ j ≤ n−1} Π ∼ = {(x1 , . . . , xn−1 ) ∈ (Rk )n−1 | x1 6= x2 } = M({U1,2 }) . b∗ associated to Π b has an E b2 -tableau of the form The spectral sequence E

b ∗,∗ E 2

k−1

b p,q ∼ E = H p (S k ) ⊗ H q (M({U1,2 })) , 2 p, q ≥ 0 .

dbk

0 0

k

b ∗,∗ we easily see that there is only From the location of non-zero entries in E 2 one possibly non-trivial differential dbk on stage k of the sequence. The inclusion of F (S k , n) into Fb is a map of fiber bundles. M( {U1,2 } ) (k)

M(An−2 )

F (S k , n)

Fb Sk

Sk The homomorphism of spectral sequences induced by the inclusion of the fiber bk -tableau into the induced map between the cohomolbundles factors on the E ogy of the fibers and the identity on the cohomology of the base space [Bo1, Exp. VIII, Thm. 4]. The map i∗ between the cohomology of the fibers maps Documenta Mathematica 5 (2000) 115–139

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(k)

the generator cb1,2 of H k−1 (M({U1,2 })) to c1,2 in H k−1 (M(An−2 )) (compare Remark 3.2). Hence, we are left to determine the action of the k-th differential b 0,k−1 : on E k Ek∗,∗

b ∗,∗ E k i∗k

c1,2

dk

b c1,2

dbk

id∗

dk (c1,2 ) = dk (i∗ (b c1,2 )) = dbk (b c1,2 ) .

Proposition 4.4 The fiber bundle Fb over S k is fiber homotopy equivalent to Vk+1,2 , the Stiefel manifold of orthogonal 2-frames in Rk+1 , considered as fiber bundle over S k . Proof. Fb is fiber homotopy equivalent to F (S k , 3), both spaces considered as fiber bundles over S k . The fiber homotopy equivalence is realized by the projection of configurations in Fb to their first, second and last points. In turn, F (S k , 3) is fiber homotopy equivalent to the Stiefel manifold Vk+1,2 [Fa, Thm. 2.4]. 2 For a simply connected, k-dimensional, orientable manifold M the only possibly non-trivial differential in the spectral sequence associated to the unit tangent bundle can be described as a cup product multiplication with the Euler class of the manifold: dk (x ⊗ µ) = dk (µ) ^ x = χM ^ x , where µ is a generator of H k−1 (S k−1 ), x ∈ H ∗ (M ), and χM denotes the Euler class of the manifold (compare [MS, Thm. 12.2]). The Stiefel manifold Vk+1,2 coincides with the unit tangent bundle on S k . Given an orientation on S k and a generator ν of H k (S k ) that evaluates to 1 on the orientation class, the Euler class of S k is given by  0 for odd k , χSk = 2ν for even k . We conclude that in the spectral sequence for Fb the differential dbk maps the generator cb1,2 of H k−1 (M({U1,2 })) to the Euler class χ of the base space, once an orientation for the base S k and with it the Euler class have been chosen Documenta Mathematica 5 (2000) 115–139

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appropriately. In particular, dbk is the zero-map for odd k. For our initial fiber bundle we thus derive  0 for odd k , dk (ci,j ) = dk (c1,2 ) = 2ν for even k , where 2ν is the Euler class of the k-sphere under appropriate orientation. 5

Recovering H ∗ (F (S k , n)) from the spectral sequence

For configuration spaces of odd-dimensional spheres we now have enough information to derive a complete description of the integer cohomology algebra. In the previous section we showed that the k-th differential is trivial on multiplicative generators of the sequence tableau Ek∗,∗ , therefore it is trivial on all of Ek∗,∗ . The spectral sequence collapses in its second term; a favorable location of non-zero tableau entries allows us to get both the linear and the multiplicative structure of H ∗ (F (S k , n)) directly from the second tableau: Theorem 5.1 For a sphere S k of odd dimension k ≥ 3, and n ≥ 3, the integer cohomology algebra of F (S k , n) is given by H ∗ (F (S k , n))

∼ = ∼ =

(k)

H ∗ (S k ) ⊗ H ∗ (M(An−2 )) M Z(k − 1) / I , ( Z(0) ⊕ Z(k) ) ⊗ Λ∗ (n−1 ) 2

where I is the ideal described in Proposition 3.1. In particular, the cohomology is free. For the case of even-dimensional spheres the considerations in the previous section show that the k-th differential is non-zero. We have to describe the kernel and cokernel of that differential and with it the final sequence tableau ∗,∗ Ek+1 in a manageable form. The cohomology algebra of the fiber, hence of the left-most column of the second, resp. k-th tableau, is given by Proposition 3.1. A linear basis for this algebra is given by the products of (k − 1)-dimensional classes ci,j associated with the faces of the broken circuit complex BC(L) of the intersection lat(k) tice L = L(An−2 ) [BZ, Sect. 9]: BBC = {cα1 ∧ . . . ∧ cαt | {α1 , . . . , αt } ∈ BC(L)} . Here is a different basis which enables us to describe the kernel of dk both as (k) a direct summand and as a subalgebra of H ∗ (M(An−2 )): (k)

Proposition 5.2 The following set is a Z-linear basis for H ∗ (M(An−2 )) : B0 = {c1,2 ∧ (cα1 − c1,2 ) ∧ . . . ∧ (cαt − c1,2 ) | {α1 , . . . , αt } ∈ BC(L), αi 6= (1, 2)} ∪ {(cα1 − c1,2 ) ∧ . . . ∧ (cαt − c1,2 ) | {α1 , . . . , αt } ∈ BC(L), αi 6= (1, 2)} . Documenta Mathematica 5 (2000) 115–139

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Proof. Each element in BBC can be written as a linear combination of elements in B 0 . This is true for each element having c1,2 as a factor because those are themselves elements in B 0 . For cα1 ∧ . . . ∧ cαt , {α1 , . . . , αt } ∈ BC(L), αi 6= (1, 2), (cα1 − c1,2 ) ∧ . . . ∧ (cαt − c1,2 ) = cα1 ∧ . . . ∧ cαt + β , where β is a linear combination of products containing c1,2 , hence of elements in B 0 . Thus cα1 ∧ . . . ∧ cαt can be written as a linear combination of those. 2 (k)

Let T • denote the submodule of H ∗ (M(An−2 )) generated by those elements of B 0 that contain c1,2 as a factor, whereas T ◦ denotes the submodule generated by all other elements of B 0 : (k) H ∗ (M(An−2 )) ∼ = T◦⊕T•.

Obviously, multiplication within T • is trivial, whereas for T ◦ we can state the following: (k)

Proposition 5.3 The submodule T ◦ is a subalgebra of H ∗ (M(An−2 )) generated by the elements c¯i,j := (ci,j − c1,2 ) in dimension k − 1, 1 ≤ i < j ≤ n − 1, (i, j) 6= (1, 2). It has a presentation as a quotient of the exterior algebra on these generators: n−1 T◦ ∼ = Λ∗ Z( 2 )−1 / J , where J is the ideal generated by elements of the form (¯ ci,l ∧ c¯j,l ) + (−1)k+1 (¯ ci,j ∧ c¯j,l ) + (¯ ci,j ∧ c¯i,l ) , 1 ≤ i < j < l ≤ n−1 , (i, j) 6= (1, 2) , (¯ c1,i ∧ c¯2,i ) , 2 < i ≤ n−1 . Proof. It is clear that T ◦ has a presentation as a quotient of the exterior algebra on the generators c¯i,j = (ci,j − c1,2 ), 1 ≤ i < j ≤ n − 1, (i, j) 6= (1, 2). (k) Moreover, it is easy to check that the proposed relations hold in H ∗ (M(An−2 )). ◦ To see that they generate the ideal for a presentation of T note that they allow one to write each product in the generators c¯i,j as a linear combination of elements from the linear basis for T ◦ : Assume that for a product of generators c¯α1 ∧ . . . ∧ c¯αt all products with lexicographically smaller index set can be written as a linear combination of basis elements from T ◦ . If this product is not itself a basis (k) element then {α1 , . . . , αt } contains a broken circuit of L(An−2 ). In case (1, 2) extends it to a circuit the product is zero by a relation of the second type. Otherwise, a relation of the first type allows to write it as a linear combination of products with lexicographically smaller index set, and hence as a linear combination of basis elements. 2 Documenta Mathematica 5 (2000) 115–139

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Our results on dk now read as follows: dk (c1,2 ) dk (ci,j − c1,2 )

= 2ν = 0

for 1 ≤ i < j ≤ n − 1 ,

where ν is a generator of H k (S k ). Evaluating dk by the Leibniz rule on the basis elements of B 0 we exhibit T ◦ as the kernel of dk , whereas im dk = 2 T ◦ , and hence coker dk ∼ = T ◦ /2T ◦ ⊕ T • . We thus obtain the final sequence ∗,∗ 0,∗ k,∗ tableau Ek+1 with entries Ek+1 = T ◦ and Ek+1 = T ◦ /2T ◦ ⊕ T • . ∗,∗ From the sequence tableau Ek+1 we can read the cohomology algebra of 0,∗ 0,0 0,k−1 F (S k , n): Free generators for T ◦ = Ek+1 are located in Ek+1 and Ek+1 . k,k−1 Together with the free generator in Ek+1 and the generator of order two in k,0 k,∗ Ek+1 they generate T ◦ /2T ◦ ⊕ T • = Ek+1 . ∗,∗ Ek+1

3k − 3

T◦

T ◦ /2T ◦ ⊕ T •

2k − 2

k−1

ci,j − c1,2

c1,2

1

ν/2ν

k−1 0

0 0

k

0

k

Linearly, the cohomology of F (S k , n) is isomorphic to a tensor product of two free generators in dimension 0 and 2k − 1 and a generator of order 2 in dimension k − 1 with the algebra T ◦ : H ∗ (F (S k , n)) ∼ = (Z(0) ⊕ Z2 (k) ⊕ Z(2k − 1)) ⊗ T ◦ . This isomorphism is an algebra isomorphism: This is obvious for multiplication 0,∗ among elements represented by entries in the left-most column Ek+1 . Also, 0,∗ k,∗ multiplication between entries of Ek+1 and Ek+1 is correctly described in the proposed tensor product. Moreover, the trivial multiplication among entries in k,∗ Ek+1 has its correspondence in the tensor algebra since multiplication within the left-hand factor is trivial. We conclude: Theorem 5.4 For a sphere S k of even dimension, k ≥ 4, the integer cohomology algebra of F (S k , n), n ≥ 3, is given by H ∗ (F (S k , n))

∼ = ∼ =

(Z(0) ⊕ Z2 (k) ⊕ Z(2k − 1)) ⊗ T ◦ (Z(0) ⊕ Z2 (k) ⊕ Z(2k − 1)) ⊗ Λ∗

M

(n−1 2 )−1 where J is the ideal described in Proposition 5.3. Documenta Mathematica 5 (2000) 115–139

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In the next section we will give a topological interpretation for this product decomposition of the cohomology algebra (see Remark 6.1). 6

A family of fiber bundles

The bundle structure on F (S k , n) given by the projection Π1 was the key to determine the integer cohomology algebra of F (S k , n). This projection Π1 is one instance from a family of fiber maps Πr = Πr (S k , n), 1 ≤ r < n, that are given by projection of a configuration in F (S k , n) to its last r points. In this section we will have a closer look at these fiber maps, at their spectral sequences, and at the question whether the induced bundle structures are trivial. For the fiber map Πr (S k , n), 1 ≤ r < n, we obtain the following space as the fiber over a point configuration q = (q1 , . . . , qr ) on S k : Π−1 r (q)

= {(x1 , . . . , xn−r ) ∈ (S k )n−r | xi 6= xj for i 6= j, xi 6= qt for i = 1, . . . , n − r, t = 1, . . . , r} .

This space is again a configuration space: k Π−1 r (q) = F ( S \ {q1 , . . . , qr }, n − r ) .

Configurations on S k that avoid r ≥ 1 (fixed) points q1 , . . . , qr are equivalent to configurations in Rk that avoid r−1 points q1 , . . . , qr−1 . Thus the fiber of Πr is homeomorphic to the complement of the arrangement AΠr (S k ,n) of (affine) subspaces in Rk(n−r) given by Ui,j Uit

= {(x1 , . . . , xn−r ) ∈ (Rk )n−r | xi = xj }, k n−r

= {(x1 , . . . , xn−r ) ∈ (R )

1 ≤ i < j ≤ n − r,

| xi = t · (1, 0, . . . , 0)T }, 1 ≤ i ≤ n − r, 0 ≤ t ≤ r − 2 .

For r = 1, the arrangement AΠ1 (S k ,n) coincides with the k-braid arrangement (k)

An−2 — a fact that we used extensively in the previous sections. For r > 2, AΠr (S k ,n) contains affine subspaces, the subspaces Uit for 0 < t ≤ r − 2. In the complex case, for k = 2, these arrangements were extensively studied by Welker [We]. 6.1

The spectral sequences

We proved in the previous sections that the spectral sequence E∗ (Π1 ) associated to the fiber map Π1 (S k , n) collapses in E2 for odd k, and in Ek+1 for even k. We obtain a similar picture for the spectral sequence E∗ (Π2 ) associated to the fiber map Π2 (S k , n): The base space F (S k , 2) is homotopy equivalent to S k . Hence, it is simply connected for k ≥ 2, and the system of local coefficients on S k induced by Π2 is simple. The fiber M(AΠ2 (S k ,n) ) is homotopy equivalent (k)

to the complement of the k-braid arrangement An−2 . In fact, the homotopy Documenta Mathematica 5 (2000) 115–139

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T (k) (k) equivalence is realized by projection of M(An−2 ) along An−2 on the linear subspace 0 Un−1 = { (x1 , . . . , xn−1 ) ∈ (Rk )n−1 | xn−1 = 0 } . Thus, the E2 -tableaux of the spectral sequences induced by Π1 and Π2 coincide. For dimensional reasons, the collapsing results on E∗ (Π1 ) translate to analogous collapsing results on E∗ (Π2 ). The picture changes for the spectral sequences E∗ (Π3 ) associated to Π3 (S k , n). In fact, we have all arguments at hand to discuss them briefly: The base space F (S k , 3) of the fiber map Π3 (S k , n) is homotopy equivalent to the Stiefel manifold Vk+1,2 of orthogonal 2-frames in Rk+1 [Fa, Thm. 2.4], hence it is simply connected for k ≥ 2. We conclude that the system of local coefficients on F (S k , 3) induced by Π3 is simple. We have seen above that the fiber of Π3 is homeomorphic to the complement of the (affine) subspace arrangement AΠ3 (S k ,n) . Comparison to the complement of the k-braid arrangement (k)

An−2 yields a homotopy equivalence, (k)

M(AΠ3 (S k ,n) ) ' M(An−2 dU ) , (k)

where An−2 dU denotes the restriction of the k-braid arrangement to the affine subspace U = {(x1 , . . . , xn−1 ) ∈ (Rk )n−1 | xn−2 − xn−1 = (1, 0, . . . , 0)T } . (k)

The homotopy equivalence is realized by projection of M(An−2 dU ) along the T (k) intersection An−2 to the linear subspace 0 Un−1 = {(x1 , . . . , xn−1 ) ∈ (Rk )n−1 | xn−1 = 0 } . (k)

The affine arrangement An−2 dU is associated to the k-braid arrangement in the same way as we associated before an affine complex hyperplane arrangement to the complex braid arrangement (compare Section 2). This analogy allows one to state a presentation for its cohomology algebra in terms of generators and relations. In fact, one obtains an algebra presentation that coincides with the one that we stated for T ◦ in Proposition 5.3: (k) H ∗ (M(An−2 dU )) ∼ = T◦. (k)

In particular, H ∗ (M(An−2 dU )) is torsion-free and it is generated in dimension k − 1 by cohomology classes that are in one-to-one correspondence with the inclusion maximal subspaces of the arrangement. For both odd and even k the E2 -tableaux of the spectral sequences associated to Π3 (S k , n) carry the structure of tensor products. We content ourselves with Documenta Mathematica 5 (2000) 115–139

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discussing the spectral sequences for k ≥ 3; for k = 2, we already showed in Section 2 that the bundle structure induced by Π3 is trivial. E2∗,∗ (Π3 ) k even

E2∗,∗ (Π3 ) k odd 3k − 3 2k − 2

3k − 3

T◦ T◦

T◦

T◦

dk

0 0

dk

k−1

dk−1

k−1

2k − 2

T ◦ ⊗ Z2

T◦

T◦

dk−1

0 k−1

k

2k−1

0

k

(k) E2p,q (Π3 ) ∼ = H p (Vk+1,2 ) ⊗ H q (M(An−2 dU )) ,

2k−1

p, q ≥ 0 .

It is easy to see that E∗ (Π3 ) collapses in its second term for both odd and even k: The location of non-zero entries in the respective tableaux suffices to see the triviality of differentials dr with r 6= k. The k-th cohomology group of F (S k , n) can be read already from the k-th diagonal in Ek+1 (Π3 ). Our results on H k (F (S k , n)) (Proposition 4.2) allow to deduce triviality of the differential dk as we did in Section 4.1. Remark 6.1 There is a topological explanation for the product decomposition of the integer cohomology algebra of F (S k , n) for even k that we obtained in Theorem 5.4: The factors are the cohomology algebras of base space and fiber for the fiber bundle structure on F (S k , n) given by Π3 . We showed above that the associated spectral sequence E∗ (Π3 ) collapses in its second term, which explains the product structure in cohomology. The collapsing result on E∗ (Π3 ) extends to the spectral sequences associated to the fiber maps Πr for r > 3, and we can summarize as follows: Proposition 6.2 The spectral sequence E∗ (Πr ) of the fiber map Πr (S k , n) on the configuration space F (S k , n) collapses in its second term unless k is even and r equals 1 or 2. For those parameters the spectral sequence collapses in Ek+1 . Proof. We are left to show the triviality of the spectral sequence E∗ (Πr ) for r > 3. This we will derive from the triviality of E∗ (Π3 ), thereby involving several applications of the following Lemma. i

Π

Lemma 6.3 [Bo2, Ch. II, Thm. 14.1] Let F ,→ E → B be a fiber bundle with path-connected base B and assume that the cohomology of the base or the cohomology of the fiber is torsion-free. Then the following assertions are equivalent: Documenta Mathematica 5 (2000) 115–139

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(1) The system of local coefficients on B induced by Π is simple and the associated spectral sequence with integer coefficients collapses in its second term. (2) The induced map i∗ : H ∗ (E) → H ∗ (F ) is surjective. Consider the map of fiber bundles between Πr (S k , n) and Π3 (S k , n) given by (id, Π3 (S k , r)). For simplicity of notation we denote with Qt a fixed set of pairwise distinct points {q1 , . . . , qt } in S k and thus write F (S k \Qt , n − t) for the respective fibers. The fibers are complements of affine k-arrangements, thus their cohomology algebras are torsion-free. F (S k \ Q3 , n − 3) i i Πr F (S k \ Qr , n − r)

i Π3

F (S k , n)

id

Π3

k

F (S , n) Πr F (S k , r)

F (S k , 3) Π3

The configuration space F (S k , 3) is simply connected for k ≥ 2, due to the homotopy equivalence with the Stiefel manifold Vk+1,2 . With the collapsing result on E∗ (Π3 ) we deduce that i∗Π3 is surjective by the equivalence stated above. We are left to show that the inclusion i between the fibers induces a surjective homomorphism in cohomology. Then i∗Πr = i∗ ◦ i∗Π3 is surjective, and another application of Lemma 6.3 yields the collapsing result on E ∗ (Πr ). To see that i∗ is surjective we interpret i as a concatenation of inclusions in a sequence of fiber maps. Namely, we consider the sequence of fiber maps obtained by successively projecting F (S k \ Q1 , n − 1) to its last coordinate. We picture the part of this sequence which is relevant to our investigation: jr−1

j

j

4 3 F (S k \Qr , n − r) −→ . . . −→ F (S k \Q4 , n − 4) −→ F (S k \Q3 , n − 3)        p4  p3   y y

S k \ Q4

S k \ Q3

The base spaces of the fiber bundles given by pt , 1 ≤ t ≤ n − 2, are simply connected for k > 2, thus the systems of local coefficients are simple. The same holds for k = 2, which is a result of Cohen [C2, Lemma 6.3]. The fibers are complements of affine k-arrangements, thus their cohomology groups are nontrivial only in dimensions that are multiples of k − 1 [GM, Part III, Thm. B]. For dimensional reasons, the associated spectral sequences E∗ (pt ) collapse in their second terms and we conclude by Lemma 6.3 that the jt∗ are surjective ∗ for 1 ≤ t ≤ n − 2. Thus, i∗ = jr−1 ◦ . . . ◦ j3∗ is a surjective map, which concludes our proof. 2

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Triviality of the fiber bundles

The fiber bundle structure induced by Π3 on F (S 2 , n) for n ≥ 3 is trivial (Theorem 2.1). One is led to ask: For which parameters do the fiber maps Πr induce a trivial fiber bundle structure on F (S k , n)? We observed in Section 2 that the bundle structure on F (S k , 2) given by Π1 is equivalent to the tangent bundle over S k . Thus, Π1 (S k , 2) is a trivial fiber map if and only if S k is parallelizable (see Hirzebruch [H]). This indicates that the triviality question for the fiber maps Πr is difficult in general. Our results on the cohomology algebra of F (S k , n) for even k, k ≥ 2, exclude a trivial bundle structure on F (S k , n) induced by Π1 : There is 2-torsion in H ∗ (F (S k , n)) while the cohomology algebra of the cartesian product of base space and fiber is torsion-free. However, the cohomology algebra of F (S k , n) for odd k coincides with the cohomology algebra of the cartesian product of base space and fiber. Such product decomposition might as well hold beyond the level of cohomology. Recall from previous arguments that F (S k , 3) is fiber homotopy equivalent to the Stiefel manifold Vk+1,2 of orthogonal 2-frames in Rk+1 , both considered as fiber bundles over S k . Fiber bundles are trivial if and only if their associated principal bundles are trivial [St, Part I, Cor. 8.4]. Hence, Vk+1,2 is a trivial fiber bundle if and only if O(k +1), considered as a fiber bundle over S k , admits a section — which again is the case iff k = 1, 3 or 7. Moreover, Vk+1,2 is fiber homotopy equivalent to a trivial bundle if and only if it is trivial itself, hence iff k = 1, 3 or 7 [Ja, Thm. 1.11]. We conclude that F (S k , 3) is a non-trivial fiber bundle over S k for k 6= 1, 3 or 7. For the 1-sphere we have shown triviality of F (S 1 , n) as a fiber bundle over S 1 in Section 2. Analogously, we obtain a trivialization of the fiber bundle structure on F (S 3 , n) given by Π1 , using the group structure of S 3 . The 7-sphere does not carry the structure of a topological group [Bd, VI, Cor. 15.21]. However, one can establish an explicit equivalence of fiber bundles between F (S 7 , 3) and V8,2 × R7 × R, both considered as fiber bundles over S 7 in the natural way. As mentioned above, V8,2 is a trivial fiber bundle over S 7 , and we can thus conclude triviality of F (S 7 , 3) over S 7 . Thus it remains to decide whether the bundle structure on F (S k , n) induced by Π1 is trivial for n > 3 and odd k ≥ 5. We have seen in Section 2 that Π3 induces a trivial bundle structure on F (S 2 , n). Our collapsing results on the spectral sequences E∗ (Π3 (S k , n)) for both odd and even k would be consistent with triviality of the fiber bundle structure given by Π3 . However, except for k = 2 this leaves us with an open question. Remark 6.4 After completion of this paper, we learned about recent work by Fadell & Husseini [FaH] which addresses the question of configuration space bundles being (fiber-homotopically) trivial. The paper is mostly concerned with configuration spaces of Euclidean spaces; a complete discussion for Documenta Mathematica 5 (2000) 115–139

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G¨ unter M. Ziegler Dept. Mathematics, MA 7-1 TU Berlin 10623 Berlin, Germany [email protected]

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